The dressing operators \(W,\bar{W}\) 
of the tau function (REF ) can be captured
by the factorization problem
(see the review [1]})
\(\exp \left(\sum _{k=1}^\infty t_k\Lambda ^k\right)U\exp \left(- \sum _{k=1}^\infty \bar{t}_k\Lambda ^{-k}\right)= W^{-1}\bar{W},\) 
